I took a look at what I can see of Rubenstein's and Neusner's books on Google books and am now wishing for a Reader's Digest version of Babylonian Jewish history. I don't think I'm ready yet for chapter titles like "From Phraates IV to Vologases I."
In the meantime, I find myself thinking about what books in the OT were written or set in Babylonia, and I suppose offhand it would start with Genesis, then 2 Kings and 2 Chronicles, Ezra, Esther (if Susa counts as part of Babylonia), Psalms, second and maybe third Isaiah, Ezekiel, Jonah and Daniel, and that seems like a significant chunk of the OT, and I wouldn't mind taking another look at them in this light.
Josephus seems like the next best source. As this "Reader's Digest" webpage notes:
The little that is known of the Jews there at the time comes from the quill of Josephus Flavius: they were very numerous and their brethren in Judea sought their help while preparing their revolt against Rome. This Roman historian also mentions two episodes which he most probably learned from literary fragments: the adventure of two brothers from Nechardea who had founded a kind of thieves-state near the city of Seleucia, and the famous conversion of the king of Adiabene to Judaism.
https://www.myjewishlearning.com/articl ... n-babylon/
This webpage goes on to say:
It is only after the fall of the Second Temple (70 C.E.) and the Bar Kokhba Revolt (132-135 C.E.) that one can truly follow the history of Babylonian Jewry, which becomes even clearer after the fall of the Parthian regime and the accession of the Sassian dynasty (224 C.E.). Sources relating to the first two centuries of the Christian era make no mention in any form of organized Torah studies in Babylonia and note hardly any Babylonian scholars. We do know that Rabbi Akiva, in his many travels, arrived in Nechardea where he announced the leap year.
After Bar Kokhba’s revolt, we hear for the first time about groups of sages who “went down” to Babylonia, undoubtedly following the religious persecution which followed the crushing of the revolt. In Babylon, the nephew of Rabbi Joshua, Hanania, attempted to proclaim the order of the Hebrew calendar, a prerogative which until then had been indisputably reserved for the leadership in Palestine. Although Hanania was forced to make a retraction, it was nevertheless the first manifestation of Babylonian independence from the Palestinian center.
But the place to start seems to be the OT.
May the four winds blow you home again.